Functional Analysis and Its Applications

, Volume 53, Issue 3, pp 157–173 | Cite as

Ultraelliptic Integrals and Two-Dimensional Sigma Functions

  • T. AyanoEmail author
  • V. M. BuchstaberEmail author


This paper is devoted to the classical problem of the inversion of ultraelliptic integrals given by basic holomorphic differentials on a curve of genus 2. Basic solutions F and G of this problem are obtained from a single-valued 4-periodic meromorphic function on the Abelian covering W of the universal hyperelliptic curve of genus 2. Here W is the nonsingular analytic curve W = {u =(u1, u3) ∈ ℂ2: σ(u) = 0}, where σ(u) is the two-dimensional sigma function. We show that G(z) = F(ξ(z)), where z is a local coordinate in a neighborhood of a point of the smooth curve W and ξ(z) is the smooth function in this neighborhood given by the equation σ(u1, ξ(u1)) = 0. We obtain differential equations for the functions F(z), G(z), and ξ(z), recurrent formulas for the coefficients of the series expansions of these functions, and a transformation of the function G(z) into the Weierstrass elliptic function ℘ under a deformation of a curve of genus 2 into an elliptic curve.

Key words

functions on sigma divisor Abel-Jacobi inversion problem 4-periodic meromorphic functions 


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The authors are grateful to A. B. Bogatyrev and V. Z. Enolski for stimulating discussions and literature references.


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Copyright information

© Springer Science+Business Media, Inc. 2019

Authors and Affiliations

  1. 1.Advanced Mathematical InstituteOsakaJapan
  2. 2.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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